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Properties

Label 2-153-153.50-c2-0-28
Degree $2$
Conductor $153$
Sign $0.242 + 0.970i$
Analytic cond. $4.16894$
Root an. cond. $2.04180$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.92 − 1.68i)2-s + (−2.55 + 1.57i)3-s + (3.70 − 6.42i)4-s + (1.55 − 2.68i)5-s + (−4.81 + 8.92i)6-s + (5.43 − 3.13i)7-s − 11.5i·8-s + (4.04 − 8.04i)9-s − 10.4i·10-s + (−1.87 − 3.25i)11-s + (0.642 + 22.2i)12-s + (1.73 − 3.01i)13-s + (10.5 − 18.3i)14-s + (0.268 + 9.30i)15-s + (−4.65 − 8.06i)16-s + (0.217 + 16.9i)17-s + ⋯
L(s)  = 1  + (1.46 − 0.844i)2-s + (−0.851 + 0.524i)3-s + (0.926 − 1.60i)4-s + (0.310 − 0.537i)5-s + (−0.802 + 1.48i)6-s + (0.776 − 0.448i)7-s − 1.44i·8-s + (0.449 − 0.893i)9-s − 1.04i·10-s + (−0.170 − 0.295i)11-s + (0.0535 + 1.85i)12-s + (0.133 − 0.231i)13-s + (0.756 − 1.31i)14-s + (0.0179 + 0.620i)15-s + (−0.291 − 0.504i)16-s + (0.0128 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(4.16894\)
Root analytic conductor: \(2.04180\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1),\ 0.242 + 0.970i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.07890 - 1.62294i\)
\(L(\frac12)\) \(\approx\) \(2.07890 - 1.62294i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.55 - 1.57i)T \)
17 \( 1 + (-0.217 - 16.9i)T \)
good2 \( 1 + (-2.92 + 1.68i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-1.55 + 2.68i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-5.43 + 3.13i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (1.87 + 3.25i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-1.73 + 3.01i)T + (-84.5 - 146. i)T^{2} \)
19 \( 1 + 18.1T + 361T^{2} \)
23 \( 1 + (0.563 - 0.976i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-22.3 - 38.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (13.2 + 7.67i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 18.6iT - 1.36e3T^{2} \)
41 \( 1 + (29.8 - 51.6i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (16.0 + 27.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (38.0 - 21.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 89.1iT - 2.80e3T^{2} \)
59 \( 1 + (-48.3 - 27.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-84.9 + 49.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-17.8 + 30.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 121.T + 5.04e3T^{2} \)
73 \( 1 - 34.4iT - 5.32e3T^{2} \)
79 \( 1 + (-91.8 + 53.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (85.2 - 49.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 59.2iT - 7.92e3T^{2} \)
97 \( 1 + (8.23 - 4.75i)T + (4.70e3 - 8.14e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.69267474718733740577554703517, −11.52658646483531143656678024114, −10.86730713322990300128728389393, −10.12667954704908523314538349747, −8.509018791527030437490317663645, −6.53570631917033509829054671348, −5.41489617391520535179375620106, −4.69714128585666782215461769373, −3.60307887674865751268288021827, −1.49664438401721473658136930849, 2.41833079158986656649956979861, 4.44764257825558652731014640484, 5.33701779584129246073652946988, 6.35978143474227670120467817699, 7.10818903791260454746898946722, 8.260291816989122957837629748042, 10.25033700488635419193566565064, 11.49697568608604574171475475030, 12.09065144337116717200326566884, 13.12941085055499619786641149808

Graph of the $Z$-function along the critical line